Question

Tìm GTNN:

a, P= (2x-1)²-(x+2)²

b, D= 2x²+2xy +2y²-6x-6y+5

Answer 1

b) \(D=2x^2+2xy+2y^2-6x-6y+5\)

\(\Leftrightarrow D=x^2+x^2+2xy+y^2+y^2-6x-6y+9+9-13\)

\(\Leftrightarrow D=\left(x^2+2xy+y^2\right)+\left(x^2-6x+9\right)+\left(y^2-6y+9\right)-13\)

\(\Leftrightarrow D=\left(x^2+2xy+y^2\right)+\left(x^2-2.x.3+3^2\right)+\left(y^2-2.y.3+3^2\right)-13\)

\(\Leftrightarrow D=\left(x+y\right)^2+\left(x-3\right)^2+\left(y-3\right)^2-13\)

Vậy GTNN của \(D=-13\) khi \(\left\{{}\begin{matrix}x+y=0\\x-3=0\\y-3=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=3\\y=3\end{matrix}\right.\)

Answer 2

\(P=\left(2x-1\right)^2-\left(x+2\right)^2\)

\(\Leftrightarrow P=\left(2x-1\right)^2-\left(x+2\right)^2+0\)

Vậy GTNN của \(P=0\) khi \(\left\{{}\begin{matrix}2x-1=0\Rightarrow x=\dfrac{1}{2}\\x+2=0\Rightarrow x=-2\end{matrix}\right.\)